2. Experimental methodology

2.1. Turbulent pipe flow

The turbulent pipe flow of drag-reducing solutions of the flexible polymer polyacrylamide (PAM) were evaluated in a recirculating flow loop shown schematically in figure 1. The selection of PAM for these experiments was due to its widespread use as a water-soluble drag reducer and its higher resistance to chain scission induced by flow shear (Pereira & Soares Reference Pereira and Soares2012; Soares et al. Reference Soares, Sandoval, Silveira, Pereira, Trevelin and Thomaz2015). Three pipe sections with different inner diameters $D$ of 26.7, 41.3 and 52.5 mm, were installed and tested within the red highlighted region of the flow loop shown in figure 1. All other piping within the flow loop had an inner diameter of 52.5 mm. Concentric reducers were used to gradually transition the pipe section to smaller diameters when required. A centrifugal pump (LCC-M 50-230, GIW Industries Inc.) was used to circulate the fluid within the loop and a Coriolis flow meter (Micro Motion F-Series, Emerson Electric Co.) was used to measure the mass flow rate $\dot {m}$ . A proportional-integral-derivative controller was used to maintain a constant $\dot {m}$ by adjusting the rotational speed of the pump using a variable frequency drive. A thermocouple and shell-and-tube heat exchanger were used to measure and maintain a constant fluid temperature of $20^{\,\circ }\text{C} \pm 0.5^{\,\circ }\text{C}$ across all experimental trials.

For each pipe diameter $D$ , five different $\dot {m}$ were considered, as listed in table 1. The bulk velocity was determined according to $U = \dot {m}/(\rho a)$ and listed in table 1, where $a = \pi D^2 /4$ is the cross-sectional area of the pipe and $\rho$ is the density. The solvent Reynolds number $Re_s$ is listed in table 1 for each pipe. The pressure drop $\Delta P$ was measured utilising a differential pressure transducer (DP-15, Validyne Engineering) along a streamwise distance of $\Delta x = 1.75$ m (corresponding to approximately $33.3D$ to $65.5D$ , depending on the pipe diameter). The most upstream pressure port was positioned 3.6 m downstream from the inlet of each pipe section (equivalent to $68.6D$ to $134.8D$ ), as illustrated in figure 1, to ensure a fully developed flow. Measurements of $\Delta P$ were collected at a sampling rate of 1 Hz using a National Instruments data acquisition system. For each $\dot {m}$ , measurements of $\Delta P$ were collected for 5 min and then averaged over time.

Table 1. List of prescribed flow conditions for each pipe section with diameter $D$ , PAM concentrations $c$ and a pure water solvent $\eta _s = 1.0$ mPa s and $\rho = 998$ kg m $^{-3}$ .

The chosen PAM polymer (6030S, SNF Floerger) had a molecular weight $M_w$ of 30–35 MDa. One concentration $c$ of PAM, equal to 50 ppm, was tested in all pipe sections with different $D$ . However, aqueous PAM solutions with larger $c$ of 100 and 150 ppm were also tested, but only in the $D = 41.3$ mm pipe. In addition to pure water with $\eta _s = 1.0$ mPa s and $\rho = 998$ kg m $^{-3}$ , two glycerol/water solutions were also examined in the $D = 26.7$ mm pipe. One solvent was a 25 % glycerol/water solution, with $\eta _s = 1.8$ mPa s and $\rho = 1089$ kg m $^{-3}$ , and the other was a 35 % glycerol/water solution, with $\eta _s = 2.5$ mPa s and $\rho = 1113$ kg m $^{-3}$ . Flow conditions for these solvents are listed in table 2. Values of $\eta _s$ were measured from steady shear rheology in a torsional rheometer. The mass flow rates $\dot {m}$ for these conditions are manipulated such that they achieve three similar $Re_s$ as those for pure water with $\eta _s = 1.0$ mPa s listed in table 1 for the same diameter of $D = 26.7$ mm. One PAM concentration of $c = 50$ ppm was tested with these more viscous glycerol/water solvents. Baseline measurements of $Re_s$ versus $C_f$ for the different solvents, without PAM, are shown in Appendix A, and all demonstrate good agreement with the Prandtl’s friction law of $C_{f}^{-1/2} = 4.0\log _{10}(Re_s C_f^{1/2})-0.4$ for Newtonian turbulent pipe flow.

Table 2. List of flow conditions for additional tests performed with more viscous solvents, the first being a 25 % glycerol/water solution and the second being a 35 % glycerol/water solution. Experiments tested here were performed in the $D = 26.7$ mm diameter pipe.

Figure 2. A sample plot of (a) mass flow rate $\dot {m}$ and (b) streamwise pressure drop $\Delta P$ versus time $t$ for a 50 ppm PAM solution degraded within the $D = 26.7\,\rm mm$ pipe. Vertical lines of similar colour represent the bounds for averaging windows, where measurements of $\dot {m}$ and $\Delta P$ were averaged.

In order to obtain solutions with lower DR, the PAM fluids were intentionally degraded by increasing the rotational speed of the pump for different amounts of time. An example of the degradation procedure for a 50 ppm PAM solution in the $D = 26.7\,\rm mm$ pipe, with a pure water solvent ( $\eta _s = 1.0$ mPa s) is shown in figure 2. Figure 2(a) shows the time series of mass flow rate $\dot {m}$ with respect to time $t$ , and figure 2(b) demonstrates the time series of pressure drop $\Delta P$ versus $t$ for the same experiment. Coloured vertical lines in figure 2 represent the bounds of different stages of the experiment. Each stage is labelled with a roman numeral in figure 2(a) that increases chronologically with $t$ . Prior to the experimental trial, a 15 l concentrated master solution was prepared. In stage (i), the master solution is added to the flow loop and circulated for 300 s at a constant pump rotational speed of 300 rpm, mixing and diluting the solution to the desired $c$ . Shortly after mixing, the rotational speed of pump is increased to 1200 rpm and the solution is intentionally degraded. This degradation phase, labelled stage (ii) and bounded by grey lines in figure 2, was 1470 s for the experiment shown in figure 2. The exact duration of stage (ii) was manipulated day to day to achieve drag-reduced flows with different $C_f$ values and polymer rheology. As an exception, to achieve $C_f$ values near the maximum DR, the solutions were not degraded, and stage (ii) was omitted. In stage (ii) of figure 2(a), polymer degradation causes $\dot {m}$ to decrease with respect to $t$ , when the pump is maintained at a constant rotational speed. Values of $\Delta P$ are large in stage (ii) and would exceed the range of the differential pressure transducer. To avoid damaging the transducer during stage (ii), the pressure sensor is isolated from the flow loop using valves; hence the appearance of $\Delta P = 0$ in stage (ii) of figure 2(b). After degradation, the rotational speed of the pump is reduced and the valves that feed the pressure transducer are re-opened. The $\dot {m}$ is then set to the lowest value of 1.26 kg s $^{-1}$ for the $D = 26.7\,\rm mm$ pipe listed in table 1 and held at that $\dot {m}$ for $\Delta t = 300$ s. The proportional-integral-derivative controller is used to manipulate the rotational speed of the pump and maintain a constant $\dot {m}$ for the 300 s duration. Stage (iii), bounded by blue lines in figure 2, is a 270 s window where $\dot {m}$ and $\Delta P$ were averaged over $t$ ; 30 s of data were neglected to avoid any transient development in the flow. The resulting average values of $\dot {m}$ and $\Delta P$ are then used in the calculation of the dimensionless numbers such as $C_f$ , $Wi$ and $Re_s$ . From stages (iv) to (vii), $\dot {m}$ was increased sequentially using the values listed in table 1 for $D = 26.7\,\rm mm$ . This procedure is the same for the $D = 41.3\,\rm $ and $D = 52.5\,\rm mm$ pipes, albeit with different prescribed values of $\dot {m}$ for stages (iii) to (vii), as listed in table 1. This degradation procedure was used to produce eight, eleven and six different degraded 50 ppm PAM solutions for the $D = 26.7\,\rm $ , 41.3 and 52.5 mm pipes, respectively. While for the 100 and 150 ppm PAM solutions, three and two different degraded solutions were considered, respectively, within the $D = 41.3\,\rm mm$ pipe. Only fresh PAM solutions (i.e. with no degradation) were tested for the two glycerol/water solvents with properties listed in table 2. Overall, this investigation considered drag-reduced pipe flows with three $D$ , five $\dot {m}$ , and 32 different PAM solutions (three $c$ , three solvents $\eta _s$ and two to eleven different amounts of degradation), thus producing a total of 156 unique trials with different rheology and $C_f$ .

2.2. Fluid rheology

For each PAM solution, and at each $\dot {m}$ , fluid samples were collected for shear and extensional rheology measurements from the flow loop half-way through the 270 s data collection intervals labelled (iii) to (vii) in figure 2. The steady shear viscosity $\eta$ with respect to shear rate $\dot {\gamma }$ was measured for each sample using a single-head torsional rheometer (DHR-2, TA Instruments) equipped with a double-gap concentric cylinder geometry. Each curve of $\eta$ versus $\dot {\gamma }$ is fit with the model of Cross (Reference Cross1965)

(2.1) \begin{equation} \frac {\eta - \eta _{\infty }}{\eta _{0}-\eta _{\infty }} = \frac {1}{1+(K \dot {\gamma })^m}, \end{equation}

where $\eta _{\infty }$ is the infinite-shear-rate viscosity, $\eta _{0}$ is the zero-shear-rate viscosity, $K$ is the consistency and $m$ is the flow index. Sample measurements of shear viscosity $\eta$ versus shear rate $\dot {\gamma }$ are shown in figure 3(a) for seven $c = 50\,\rm ppm$ PAM solutions with different amounts of degradation. At low $\dot {\gamma }$ , measurements of $\eta$ are neglected due to the low torque limitation ( $M \lt 600$ nN m) of the device, as indicated by the solid black line on the left side of figure 3(a). At larger $\dot {\gamma }$ , measurements of $\eta$ demonstrate an abrupt increase due to the inception of Taylor vortices at a Taylor number $Ta$ of 1700 and indicated by the solid black line on the right side of figure 3(a). The coloured lines represent Cross model (Cross Reference Cross1965) fits represented by (2.1). Figure 3(a) demonstrates that as the 50 ppm PAM solutions degrade, $\eta$ decreases for all $\dot {\gamma }$ . Furthermore, $\eta _0$ demonstrates a notable reduction with increasing amounts of degradation; therefore $\beta = \eta _s/\eta _0$ should increase as PAM solutions degrade. In this case, for the given 50 ppm PAM solutions, $\beta$ increased from 0.56 to 0.87 with increasing amounts of degradation. For the 100 ppm PAM solutions $\beta$ was between 0.5 and 0.7, while the 150 ppm PAM solutions had $\beta$ between 0.2 and 0.5.

Figure 3. Plots of (a) steady shear viscosity with respect to shear rate and (b) filament diameter versus time from liquid dripping extensional rheometry. Solid lines in (a) are fits using (2.1). Solid lines in (b) are fits using (2.2).

Considering the polymer solutions are shear thinning, with shear rheograms described well by (2.1), some doubt is cast on the validity of the assumptions used in § 1 to arrive at $\psi$ of (1.1). Given the polymers are shear thinning, questions arise as to why $K$ , $m$ or $\eta _{\infty }$ are not included in (1.1). Although shear thinning has been shown to cause DR (Singh, Rudman & Blackburn Reference Singh, Rudman and Blackburn2017; Arosemena et al. Reference Arosemena, Andersson, Andersson and Solsvik2021), its contributions to DR are assumed to be minimal compared with viscoelasticity. Drag reduction using a inelastic Carreau shear thinning constitutive models was investigated using turbulent channel flow direct numerical simulation (DNS) by Arosemena et al. (Reference Arosemena, Andersson, Andersson and Solsvik2021). Despite large amounts of shear thinning, with $\eta _{\infty }/\eta _0 = 1\times 10^{-3}$ , Arosemena et al. (Reference Arosemena, Andersson, Andersson and Solsvik2021) reported only a small DR percentage of 10 %. Given the smallest $\eta _{\infty }/\eta _0$ among all of the present drag-reducing PAM solutions is 0.36, which is much closer to unity than that used by Arosemena et al. (Reference Arosemena, Andersson, Andersson and Solsvik2021), it is expected that shear thinning does not contribute significantly to $C_f$ in these fluids.

Liquid dripping extensional rheology was used to determine $\lambda$ for each PAM solution (Deblais et al. Reference Deblais, Herrada, Eggers and Bonn2020; Rajesh, Thiévenaz & Sauret Reference Rajesh, Thiévenaz and Sauret2022). Here, a droplet of each PAM solution is expelled from a blunt end nozzle with an outer diameter of $D_0= 1.27\,\rm mm$ , using a syringe pump with a flow rate of 0.02 ml min $^{-1}$ . As the droplet is expelled from the nozzle, a liquid bridge forms between the droplet and the outlet of the nozzle. A high-speed camera (NOVA S9, Photron Inc.) and light emitting diode were used to record images of the minimum diameter $D_{min }$ of the liquid bridge with respect to time $t$ . Viscoelastic polymer solutions tend to exhibit elasto-capillary (EC) thinning, where $D_{min}/D_0$ closely resembles an exponential function of the form

(2.2) \begin{equation} \frac {D_{min}}{D_0} = C \exp \left (-\frac {t-t_b}{3 \lambda }\right ). \end{equation}

Here, $C$ is a fitting constant, $t_b$ is an inertial break-up time and $\lambda$ is the elastic relaxation time (Anna & McKinley Reference Anna and McKinley2001). For each sample, extensional rheology measurements were repeated three times. The $\lambda$ of each solution was taken to be the average of the three values of $\lambda$ derived from the fits of (2.2). Sample measurements of extensional rheology, given by $D_{min}/D_0$ with respect to $t-t_b$ , are shown in figure 3(b) for the same PAM solutions shown in figure 3(a) with different amounts of degradation. For $t\gt t_b$ the PAM solutions exhibit EC thinning, where $D_{min}/D_0$ resembles (2.2). In the EC regime, $\lambda$ tends to be smaller for PAM solutions that experience larger amounts of degradation. In this case, the 50 ppm solutions shown in figure 3(b) have a $\lambda$ that reduces from 5.2 to 0.19 ms with increasing amounts of degradation. Regardless of the concentration $c$ , PAM solutions that experience more mechanical degradation tend to have lower values of $\lambda$ , similar to that shown previously in Owolabi et al. (Reference Owolabi, Dennis and Poole2017) using CaBER measurements of $\lambda$ . That said, $\lambda$ tends to be overall larger for solutions with greater $c$ .

The uncertainty in the measurements of $\dot {m}$ and $\Delta P$ consist of systematic uncertainties, prescribed by the sensor manufacturer, and random uncertainties, established from the repeatability in the measurements of $\dot {m}$ and $\Delta P$ for water. The relative uncertainty in $\eta _0$ was conservatively assessed to be $\pm 4.0\,\%$ for all PAM solutions. Uncertainty in $\lambda$ was assumed to be the range in the thrice-repeated measurements of $\lambda$ from liquid dripping extensional rheology. Using these estimates, the uncertainty in the dimensionless variables was assessed from propagation of uncertainty (Wheeler & Ganji Reference Wheeler and Ganji2010), and are represented by error bars in all figures to follow. A detailed discussion of the measurement uncertainties is provided in Appendix A.

Figure 4. The skin-friction coefficient as a function of solvent viscosity Reynolds number. The upper and lower black lines are the $C_f$ for Newtonian turbulent pipe flow and MDR (Virk et al. Reference Virk, Mickley and Smith1970), respectively.

Figure 5. Plots of (a) $\textit{DR}$ versus $Wi \beta$ and (b) $A$ versus $Wi \beta$ for the polymeric flows with different $D$ and $c$ . The black lines in (a,b) represent the fit functions labelled on each plot. Marker symbols and colours are the same as those shown in figure 4(a).

3. Skin-friction scaling

Plots of $C_f$ versus $Re_s$ are shown in figure 4(a) for all 156 pipe flow measurements of PAM with different $D$ , $\dot {m}$ , $c$ , $\eta _s$ and degradation conditions. Markers are coloured according to $D$ , while marker symbols correspond to different $c$ and $\eta _s$ as indicated by the legend above figure 4(a). Figure 4(a) demonstrates that the measurements cover a wide range of $C_f$ values, that fall between the $C_f$ versus $Re$ distributions for Newtonian turbulent pipe flows according to Prandtl’s friction law of $C_{f}^{-1/2} = 4.0\log _{10}(Re_s C_f^{1/2})-0.4$ and maximum drag reduction (MDR) according to $C_{f}^{-1/2} = 19.0\log _{10}(Re_s C_f^{1/2})-32.4$ (Virk, Mickley & Smith Reference Virk, Mickley and Smith1970). As the PAM solutions are degraded, $C_f$ increases for a given $Re_s$ , as annotated in figure 4(a). Note, that typically $Re_s$ is not used in plots of $C_f$ for polymer drag-reduced flows; rather, a Reynolds number based on a near-wall viscosity $Re_w = \rho U D/\eta _w$ is used (Xi Reference Xi2019), where $\eta _w$ is the near-wall viscosity. Different definitions of the Reynolds number, including $Re_w$ and $Re_0 = \rho U D / \eta _0$ , are explored later in § 4 of the manuscript.

Figure 4(b) shows a plot of $C_f$ versus $Re_s$ , with markers coloured according to each fluid’s respective value of $Wi \beta$ . The $Wi$ represents the extent to which the polymers are stretched based on the bulk shear rate $U/D$ . In general, a higher $Wi$ indicates greater polymer stretching, which is expected to result in a lower $C_f$ , establishing an inverse relationship. However, this approximation of $Wi$ , derived in § 1, does not account for the effect of polymers on viscosity. As inferred from figure 3, PAM solutions with higher $\lambda$ (and thus higher $Wi$ ) exhibit greater viscosity, which leads to a smaller $\beta$ . Since this smaller $\beta$ contributes to a larger $C_f$ , which is also an inverse relationship, $Wi$ is multiplied by $\beta$ to compensate for the viscosity effect. This is confirmed in figure 4(b) showing that $C_f$ decreases as $Wi \beta$ increases for a given $Re_s$ . Flows with small values of $Wi \beta$ , close to zero, have values of $C_f$ close to the Newtonian skin-friction correlation, while $Wi \beta$ in excess of 0.2 produce values of $C_f$ that approach MDR. We acknowledge that use of $Wi \beta$ is in contradiction with the prior work of Chandra et al. (Reference Chandra, Shankar and Das2020), who demonstrated that $C_f$ scales with $Wi(1-\beta )$ . Scaling of $C_f$ with $Wi(1-\beta )$ is discussed in greater detail in § 5.

A DR percentage $\textit{DR}$ is often used to estimate the reduction in $C_f$ of each polymer solution relative to Newtonian turbulence (Owolabi et al. Reference Owolabi, Dennis and Poole2017; Xi Reference Xi2019). Here, the DR percentage of each polymer drag-reduced pipe flow is defined by the equation,

(3.1) \begin{equation} DR = \left (1- \frac {C_{f}}{C_{f,N}} \right ) \times 100\,\%, \end{equation}

where $C_{f,N}$ is a Newtonian skin-friction coefficient at a similar $Re_s$ (or $\dot {m}$ ) following Prandtl’s friction law. Uncertainty in $\textit{DR}$ propagates from the errors in $C_f$ , discussed in Appendix A, and is represented by error bars in plots of $\textit{DR}$ that follow. Other definitions of $\textit{DR}$ , using $C_{f,N}$ at a similar $Re_0$ and $Re_w$ as the polymeric flows, are also explored later in § 5.

Figure 5(a) demonstrates $\textit{DR}$ as a function of $Wi \beta$ for all polymer solutions with different $c$ and $D$ . Values of $\textit{DR}$ show good consistency with one another when plotted with respect to $Wi \beta$ , and are well represented by an inverse relationship $\widehat {DR} = 79.3\,\%[-0.0173(Wi \beta +0.0143)^{-1}+1]$ . Here, the hat symbol $\widehat {\cdots }$ is used to represent a prediction based on a fit of the measurements. This inverse relationship was chosen because it predicts an onset DR $(\textit{DR}=0\%)$ , and an asymptotic value of $\textit{DR}$ , corresponding roughly to the limit of MDR in the range of $Re_s$ considered. Based on the fit of $\widehat {DR}$ , DR commences when $Wi \beta$ exceeds $3.00 \times 10^{-3}$ , and $\textit{DR}$ asymptotically saturates to 79.3 %. The equation for $\widehat {DR}$ fits the measurements well, given a coefficient of determination $R^2 = 1 - \sum _{n=1}^{N}(DR_n-\widehat {DR}_n)^2/\sum_{n=1}^N(DR_n-\langle DR \rangle )^2$ of 0.954. Here, $n$ is a sample measurement, $N = 156$ is the total number of measurements and angle brackets $\langle \cdots\kern-0.3pt \rangle$ represent the average of the quantity contained within the brackets.

Although $\textit{DR}$ shows good scaling with $Wi \beta$ , the proposed inverse equation has some limitations. Notably, the asymptotic limit of $\widehat {DR}$ corresponding to MDR is only expected to apply for the range of $Re_s$ measured here. This is because the skin-friction equation for Newtonian turbulence and MDR diverge with increasing $Re_s$ , and the value of $\textit{DR}$ corresponding to MDR increases with $Re_s$ . To alleviate this concern, a new variable $A$ is used to describe the proximity of $C_f$ for each drag-reduced flow to the skin-friction coefficient for Newtonian turbulence and MDR. Here, $A$ is the slope of $C_f$ and $Re_s$ in Prandtl–von Kármán coordinates according to

(3.2a) \begin{align}C_f^{-1/2} &= A \log _{10}(Re_sC_f^{1/2}) + B,\end{align}

(3.2b) \begin{align}B &= \frac {122}{15}-\frac {32A}{15}, \end{align}

where $B$ is the intercept of $C_f^{-1/2}$ at $Re_sC_f^{1/2} = 1$ . Note that, when $A = 4.0$ and 19.0, (3.2) produces the skin-friction correlations for Newtonian turbulence and MDR, respectively. Here $B$ , represented by (3.2b ), is defined such that all values of $A$ produce lines of $C_f$ versus $Re_s$ that pass through the point of intersection between the lines for Newtonian turbulence and MDR. Values of $A$ are determined for each flow based on their respective value of $C_f$ and $Re_s$ and from rearranging (3.2), as $A = (C_f^{-1/2}-122/15)/(\log _{10}(Re_sC_f^{1/2})-32/15)$ . Therefore, each measured value of $C_f$ and $Re_s$ , shown in figure 4(a), has a corresponding value of $A$ based on (3.2). Considering all values of $C_f$ for the polymer drag-reduced flows are less than Prandtl’s friction law, but greater than MDR – as demonstrated in figure 4(a) – it is expected that $A$ will be bounded between 4.0 and 19.0.

A plot of $A$ versus $Wi \beta$ for each flow with known $C_f$ and $Re_s$ values, is shown in figure 5(b). As predicted, values of $A$ are between 4.0 and 19.0, corresponding to Newtonian turbulence and MDR, respectively. When plotted with respect to $Wi \beta$ , values of $A$ show good consistency with one another for the flows of different $D$ and $c$ , and are accurately described by the power-law function $\widehat {A} = 27.6(Wi \beta )^{0.346}$ . The $R^2$ of this fit is 0.949, which is comparable to that of $\widehat {DR}$ . However, unlike the inverse function of $\widehat {DR}$ , the function $\widehat {A}$ enables a more obvious prediction of DR onset and MDR, which would occur where $\widehat {A} = 4$ and 19, respectively. Based on the power-law function of $\widehat {A}$ versus $Wi \beta$ , DR onset would occur at $Wi \beta = 3.76 \times 10^{-3}$ , not too dissimilar from the $Wi \beta = 3.00 \times 10^{-3}$ predicted by $\widehat {DR}$ , and MDR would occur at $Wi \beta = 3.40 \times 10^{-1}$ . Substituting $\widehat {A}$ for $A$ in (3.2), provides an equation for the skin-friction coefficient as a function of $Re_s$ , $Wi$ and $\beta$ ; in other words, the function $\psi$ , as per (1.1). The function is represented as

(3.3a) \begin{align} \widehat {C_f}^{-1/2} = \widehat {A} \log _{10}(Re_s\widehat {C_f}^{1/2}) + \widehat {B}, \end{align}

where

(3.3b) \begin{align} \widehat {A} = 27.6(Wi \beta )^{0.346}, \end{align}

and

(3.3c) \begin{align} \widehat {B} = 122/15-58.9(Wi \beta )^{0.346}, \end{align}

and $\widehat {C_f}$ represents the predicted value of skin-friction coefficient based on $Re_s$ , $Wi$ and $\beta$ . As a final simplification, the implicit function of (3.3a ) can be made explicit through use of the zeroth branch of the Lambert function (Goudar & Sonnad Reference Goudar and Sonnad2003). Therefore, (3.3a ) can be equally represented as

(3.4) \begin{equation} \widehat {C_f}^{-1/2} = \frac {\widehat {A}}{\ln (10)} \mathcal{W}\left[\frac {Re_s \ln (10)}{\widehat {A}} \exp \left (\frac {\widehat {B} \ln (10)}{\widehat {A}} \right ) \right], \end{equation}

where $\mathcal{W}$ is the Lambert function, which is the inverse operation of $f(w) = w \exp (w)$ . Here, $w$ is an arbitrary dependent variable. In other words, the Lambert function can be expressed as, $w = \mathcal{W}[w \exp (w)]$ .

A per cent difference between $\widehat {C_f}$ and $C_f$ , equal to $\omega = (\widehat {C_f} - C_f)/C_f \times 100\,\%$ , is also used to evaluate the accuracy of (3.3). Figure 6(a) demonstrates $C_f$ versus $Re_s$ , similar to figure 4(a), however, with markers coloured according to values of $\omega$ . Large values of $\omega$ appear to typically occur at lower $C_f$ or larger $\textit{DR}$ and $Wi\beta$ . One condition in particular with $C_f$ between $1\times 10^{-3}$ and $2\times 10^{-3}$ , corresponding to the flow with $D = 41.3\,\rm mm$ and $c = 50\,\rm ppm$ , shows values of $\omega$ near 20 % for all $Re_s$ , perhaps highlighting an anomalous test day. The probability density function (p.d.f.) of $\omega$ is shown in figure 6(b). The p.d.f. in $\omega$ appears to be slightly skewed to more positive values with a median of −0.332 % and a mean of 0.318 %. Values of $\omega$ vary between $\pm 20$ %, as demonstrated by the extents of the p.d.f. shown in figure 6(b). An approximate estimate for the accuracy of the prediction (3.3) is assumed to be the standard deviation in $\omega$ , which is 9.36 %.

Figure 6. Plot of (a) $C_f$ versus $Re_s$ with lines and markers coloured based on $A$ and $\widehat {A}$ respectively, and (b) p.d.f. of the per cent difference between $\widehat {C_f}$ and $C_f$ , or $\omega$ , for each flow.

4. Reynolds number definition

Use of $Re_s$ is not typical when considering polymer drag-reduced pipe flows (Xi Reference Xi2019); generally, the enhanced viscosity of the polymer solutions needs to be taken into account. For that reason, the influence of different Reynolds number definitions are discussed in this section. Figure 7 demonstrates the skin-friction scaling for the two other definitions of the Reynolds number; $Re_0 = \rho U D/\eta _0$ and $Re_w = \rho U D/\eta _w$ . Marker colours and symbols, that represent the polymer drag-reduced flows of different $D$ and $c$ , are the same as those defined earlier in figure 4(a).

Figure 7(a) demonstrate $C_f$ versus $Re_0$ . Considering $\eta _0 \gt \eta _s$ , values of $Re_0$ are lower than $Re_s$ shown in figure 4(a). The PAM solutions with $c =100\,\rm $ and $c = 150\,\rm ppm $ have lower $Re_0$ values than most of the $c = 50\,\rm ppm$ flows, due to their much larger $\eta _0$ . In some instances, values of $C_f$ and $Re_0$ for the 150 ppm solution fall slightly below the MDR asymptote, calculated based on $Re_0$ . Figure 7(b) demonstrates the DR percentage of the polymer drag-reduced flows evaluated according to (3.1), but with respect to a Newtonian skin-friction coefficient $C_{f,N}$ at a similar $Re_0$ . This DR percentage is defined as $\textit{DR}_0$ . Despite the difference between $\textit{DR}$ and $\textit{DR}_0$ , values of $\textit{DR}_0$ appear to follow a similar inverse relationship; in this case $\textit{DR}_0$ is well predicted by the function $\widehat {DR}_0 = 82.9\,\%[-0.0167(Wi \beta +0.0142)^{-1} + 1]$ . The coefficient of determination $R^2$ of the function $\widehat {DR}_0$ is 0.958, which is slightly better than that of $\widehat {DR}$ . Figure 7(c) demonstrates the parameter $A_0$ , defined from rearranging (3.2), but with $Re_0$ in lieu of $Re_s$ . Similar to $A$ , values of $A_0$ faithfully follow a power-law trend with respect to $Wi \beta$ , but represented by $\widehat {A}_0 = 37.0(Wi \beta )^{0.398}$ . The prediction $\widehat {A}_0$ has an $R^2 = 0.926$ that is marginally worse than the $R^2$ of 0.949 for $\widehat {A}$ . Certain values of $A_0$ also exceed 19 (corresponding to MDR), particularly at larger $c$ and lower $D$ , hinting that $Re_0$ is not an ideal Reynolds number definition for the flows.

Figure 7. Plots of (a) $C_f$ versus zero-shear-viscosity Reynolds number $Re_0$ , (b) $\textit{DR}_0$ based on similar $Re_0$ versus $Wi \beta$ , (c) slope factor $A_0$ versus $Wi \beta$ , (d) $C_f$ versus wall-viscosity Reynolds number $Re_w$ , (e) $\textit{DR}_w$ based on similar $Re_w$ versus $Wi \beta$ , (f) slope factor $A_w$ versus $Wi \beta$ . Lines in (a,d) labelled ‘NT’ correspond to Newtonian turbulence $C_f^{-1/2} = 4.0\log _{10}(Re C_f^{1/2})-0.4$ and ‘MDR’ correspond to maximum DR $C_f^{-1/2} = 19.0\log _{10}(Re C_f^{1/2})-32.4$ (Virk et al. Reference Virk, Mickley and Smith1970), where $Re = Re_0$ in (a) and $Re=Re_w$ in (d). Black solid lines in (b,c) and (e,f) correspond to the equations annotated on each respective panel. Marker symbols and colours are the same as those labelled in figure 4(a).

Figure 7(d) shows $C_f$ versus a Reynolds number $Re_w$ based on the near-wall viscosity $\eta _w$ . Here, $\eta _w$ is derived using the Cross model (2.1) coupled with measurements of the wall shear stress $\tau _w = (D/4)\Delta P / \Delta x$ , which can be equally represented as $\tau _w = \eta _w \dot {\gamma }_w$ , where $\dot {\gamma }_w$ is the near-wall shear rate. This Reynolds number definition $Re_w$ is typically used in most investigations of polymer drag-reduced flows and is therefore, important to consider (Xi Reference Xi2019). Generally, $\eta _0 \gt \eta _w \gt \eta _s$ and $Re_w$ tends to be lower than $Re_s$ , but larger than values of $Re_0$ – as seen when comparing figure 4(a) with figures 7(a) and 7(d). The DR percentage $\textit{DR}_w$ and the slope parameter $A_w$ based on $Re_w$ , shown in figure 7(e,f), respectively, demonstrate similar trends as those seen for $\textit{DR}$ and $A$ , with subtle differences in value and the quality of fit. For example, the function $\widehat {A}_w$ has a better data collapse than $\widehat {A}$ and $\widehat {A}_0$ , with an $R^2$ of 0.964, implying $Re_w$ is the most appropriate Reynolds number definition. That said, for the sake of predicting DR or $C_f$ , $Re_s$ is suitable; since the wall viscosity $\eta _w$ is not known a priori.

5. Previous correlations of drag reduction and viscoelastic rheology

The most relevant rheological correlation of $\textit{DR}$ and elastic flow features was performed by Owolabi et al. (Reference Owolabi, Dennis and Poole2017). As discussed in § 1, Owolabi et al. (Reference Owolabi, Dennis and Poole2017) correlated $\textit{DR}$ with a near-wall Weissenberg number $Wi_{\tau } = \lambda \dot {\gamma }_w$ , where $\dot {\gamma }_w$ is a wall shear rate estimated from measurements of $\tau _w$ from $\Delta P$ and shear rheograms. Owolabi et al. (Reference Owolabi, Dennis and Poole2017) proposed that $\textit{DR}$ should scale with $Wi_{\tau }$ according to

(5.1) \begin{equation} DR = 2C_1\left (\frac {1}{1+\exp (Wi_c-Wi_{\tau })} - Wi_c \right ), \end{equation}

where $C_1$ is the asymptotic limit of $\textit{DR}$ as $Wi_{\tau } \rightarrow \infty$ and $Wi_c$ is the critical onset value $Wi_{\tau }$ where $\textit{DR}$ starts to exceed 0 %. Owolabi et al. (Reference Owolabi, Dennis and Poole2017) based (5.1) on a trend derived by Housiadas & Beris (Reference Housiadas and Beris2013) that was established using DNS with different viscoelastic constitutive models (Oldroyd-B, Giesekus and FENE-P [finitely extensible non-linear elastic dumbells with a Peterlin approximation]). The trend established by Housiadas & Beris (Reference Housiadas and Beris2013) is

(5.2) \begin{equation} DR = C_1\left (1+\frac {2}{1+\exp \Big(\frac {Wi_{\tau }-Wi_c}{\sigma }\Big)} \right ), \end{equation}

where $\sigma$ is a fitting constant that modifies the rate at which $\textit{DR}$ increases with $Wi_{\tau }$ . Note that, (5.1) from Owolabi et al. (Reference Owolabi, Dennis and Poole2017) is the same as (5.2) from Housiadas & Beris (Reference Housiadas and Beris2013), for $\sigma = 1$ and $Wi_c = 0.5$ . Figure 8 demonstrates $\textit{DR}$ as a function of the near-wall Weissenberg number $Wi_{\tau }$ for the present polymer drag-reduced flows. Equation (5.1) is shown in figure 8 using a dashed black line for $C_1 = 79.3$ %, which corresponds to MDR as shown in figure 5(a), and $Wi_c = 0.5$ from Owolabi et al. (Reference Owolabi, Dennis and Poole2017). Measurements of $\textit{DR}$ increase with growing $Wi_{\tau }$ , but do not show a similar collapse on the profile of (5.1) previously seen by Owolabi et al. (Reference Owolabi, Dennis and Poole2017). The onset $Wi_{\tau }$ of 0.5 appears to be consistent with Owolabi et al. (Reference Owolabi, Dennis and Poole2017), but for all $Wi_{\tau } \gt Wi_c$ measurements deviate from (5.1). The coefficient of determination $R^2$ for (5.1) relative to the current measurements is 0.24, which is much worse than the $R^2$ of 0.820 observed by Owolabi et al. (Reference Owolabi, Dennis and Poole2017).

The model of (5.1) can be improved by incorporating an additional fitting parameter ( $\sigma$ ) and utilising the trend of (5.2) proposed by Housiadas & Beris (Reference Housiadas and Beris2013). Using the same onset Weissenberg number $Wi_c = 0.5$ and asymptotic DR percentage $C_1 = 79.3$ % used for (5.1), but with $\sigma = 2$ , the trend in $\textit{DR}$ versus $Wi_{\tau }$ is much more accurately modelled, with an $R^2$ of 0.76. Recall that the model of Owolabi et al. (Reference Owolabi, Dennis and Poole2017) corresponds to $\sigma = 1$ ; therefore, comparing the dashed and solid black lines corresponding to (5.1) and (5.2), respectively, demonstrates that increasing $\sigma$ in (5.2) will delay the transition to maximum DR $C_1$ or reduce the rate at which $\textit{DR}$ increases with growing $Wi_{\tau }$ . Interestingly, much of the deviation between (5.2) corresponds to flows with larger $c$ , despite the fact that the fluids with larger $c$ have a more comparable $\beta$ to the PAM solutions with $\beta \approx 0.1$ used in Owolabi et al. (Reference Owolabi, Dennis and Poole2017).

One potential cause for a large error in figure 8 could be attributed to a large uncertainty in measuring $\dot {\gamma }_w$ for calculating $Wi_{\tau }$ . The maximum measurable shear rate $\dot {\gamma }$ in the given rheometer using the double gap Taylor–Couette geometry was approximately 200 s $^{-1}$ . However, near-wall shear rates in the turbulent pipe flow are on the order of $10^3$ to $10^4$ s $^{-1}$ , particularly for the smallest $D$ of 26.7 mm. Extrapolating shear rheograms using shear thinning models, such as the model of Cross (Reference Cross1965) and given by (2.1), are typically known to produce large uncertainties in $\eta _w$ and $\dot {\gamma }_w$ (Singh et al. Reference Singh, Rudman, Blackburn, Chryss, Pullum and Graham2016). Measurements of $\eta$ at larger $\dot {\gamma }$ would provide better confidence in the resulting values of $Wi_{\tau }$ and comparison with (5.1); however, it is unclear if shear rheograms were also extrapolated in the work of Owolabi et al. (Reference Owolabi, Dennis and Poole2017) in their derivation of $\dot {\gamma }_w$ .

Figure 8. Drag reduction percentage based on constant $\dot {m}$ or $Re_s$ with respect to near-wall Weissenberg number $Wi_{\tau }$ . The dashed black line represents the correlation of $\textit{DR}$ and $Wi_{\tau }$ (5.1) from Owolabi et al. (Reference Owolabi, Dennis and Poole2017) with $C_1 = 79.3\,\%$ and $Wi_c = 0.5$ . The solid black line represents the model (5.2) from Housiadas & Beris (Reference Housiadas and Beris2013) with an additional fitting parameter of $\sigma = 2$ . Marker symbols and colours are the same as those presented throughout the manuscript and labelled in figure 4(a).

Figure 9. Drag reduction percentage based on constant $\dot {m}$ or $Re_s$ with respect to (a) $Wi$ , (b) $1-\beta$ and (c) $Wi(1-\beta )$ . Marker symbols and colours are the same as those labelled in figures 4(a) and 8.

Another work that demonstrated scaling of $C_f$ and $\textit{DR}$ with viscoelastic rheology was that of Chandra et al. (Reference Chandra, Shankar and Das2020). In their measurements of $C_f$ in polymer drag-reduced microtubes, Chandra et al. (Reference Chandra, Shankar and Das2020) demonstrated good scaling between $\textit{DR}$ and $Wi(1-\beta )$ . In the present investigation, better scaling was observed between $\textit{DR}$ and $Wi \beta$ , and not $Wi(1-\beta )$ , as shown in figure 5(a). The following discussion using figure 9 shows the discrepancy that exists if $Wi(1-\beta )$ is used.

Figure 9(a) demonstrates $\textit{DR}$ versus $Wi$ for all flows. Values of $\textit{DR}$ versus $Wi$ for different $D$ , but a similar $c$ of 50 ppm, tend to overlap; however, for a given $\textit{DR}$ (e.g. $\textit{DR} = 50\,\%$ ) and $D$ of 41.3 mm, $Wi$ increases as $c$ grows. Values of $\lambda$ are known to increase with $c$ (Rajesh et al. Reference Rajesh, Thiévenaz and Sauret2022), hence $Wi = \lambda U/D$ is expected to also increase with $c$ for a given $U$ and $D$ . Similarly, for a given $\textit{DR}$ , $1-\beta$ also tends to increase with $c$ , as demonstrated by the plot of $\textit{DR}$ versus $1-\beta$ of figure 9(b). As shown in figure 9(c), multiplying $Wi$ and $1-\beta$ does not collapse the data, as previously demonstrated by Chandra et al. (Reference Chandra, Shankar and Das2020), but rather, amplifies the discrepancies between the different PAM concentrations. On the other hand, multiplying $Wi$ with $\beta$ tempers the spread in the measurements of $\textit{DR}$ for different $c$ , as shown in figure 5(a). At a given $\textit{DR}$ , values of $\beta$ are smaller for larger $c$ , counteracting the spread in $Wi$ due to concentration differences shown in figure 9(a). It should be noted that the present experiments are different than those of Chandra et al. (Reference Chandra, Shankar and Das2020), who considered polymer drag-reduced flows with different $c$ and polymer type, but not degraded polymer solutions. Furthermore, Chandra et al. (Reference Chandra, Shankar and Das2020) does not explicitly determine $\lambda$ from extensional viscosity measurements, but utilises a correlation of $\lambda$ and $c$ .

6. Further discussion

Expanding upon the discussion of figure 9, we elaborate further on the use of $Wi \beta$ as a combination of scaling parameters in the following discussion. Recall that $Wi = U \lambda /D$ represents the extent polymers are stretched within the flow (based on a bulk shear rate $U/D$ ), which is an increasing function of $c$ . On the other hand, $\beta = \eta _s/\eta _0$ is meant to account for the influence of polymers on shear viscosity, which is a decreasing function of $c$ . Therefore, the combination $Wi\beta$ can be interpreted as an ‘effective Weissenberg number’, or a viscosity corrected bulk Weissenberg number. This definition of the effective Weissenberg number ( $Wi\beta$ ) accounts for differences in $c$ better than other effective Weissenberg number definitions, e.g. $Wi(1-\beta )$ , when predicting $C_f$ or $\textit{DR}$ , as demonstrated when comparing figure 5(a) with 9(c).

Although the proposed model of (3.3) is able to accurately predict $C_f$ , as shown by figure 6, its derivation is predicated on the assumption that viscoelasticity is the main contributor to DR, as stated in § 1. That said, it is possible other rheological complexities, not measured in the present investigation, could contribute to skin-friction drag or be used in lieu of $\lambda$ for prediction of $C_f$ . For example, previous investigations have demonstrated that the first normal stress differences $N_1$ roughly scales with DR (Escudier, Presti & Smith Reference Escudier, Presti and Smith1999); however, values of $N_1$ were also previously shown to be correlated with $\lambda$ and might not imply a distinct mechanism for $\textit{DR}$ (Zell et al. Reference Zell, Gier, Rafai and Wagner2010). Another rheological property that could have been considered was the extensional viscosity $\eta _{ext}$ , which is widely regarded as a relevant mechanism for DR (Lumley Reference Lumley1973). In dimensionless terms, the ratio of the extensional to shear viscosity is the Trouton ratio $Tr = \eta _{ext}/\eta$ . Measurements of $\eta _{ext}$ could be implied from the present extensional rheology measurements, although strain rates are likely too low for comparison with the turbulent pipe flows. Alternatively, the recently developed stagnation-point extensional rheology techniques could be used to derive measurements of the steady state extensional viscosity $\eta _{ext}$ (Haward et al. Reference Haward, Pimenta, Varchanis, Carlson, Toda-Peters, Alves and Shen2023a ,Reference Haward, Varchanis, McKinley, Alves and Shen b ).